Unit #11 : Integration by Parts, Average of a Function. Goals: Learning integration by parts. Computing the average value of a function.

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1 Unit #11 : Integration by Parts, Average of a Function Goals: Learning integration by parts. Computing the average value of a function.

2 Integration Method - By Parts - 1 Integration by Parts So far in studying integrals we have used direct anti-differentiation, for relatively simple functions, and integration by substitution, for some more complex integrals. However, there are many integrals that can t be evaluated with these techniques. Try to find xe 4x dx.

3 Integration Method - By Parts - 2 This particular integral can be evaluated with a different integration technique, integration by parts. This rule is related to the product rule for derivatives. Expand d dx (uv) = Integrate both sides with respect to x and simplify. Express u dv dx dx relative to the other terms.

4 Integration Method - By Parts - 3 Integration by Parts For short, we can remember this formula as udv = uv vdu Integration by parts: Choose a part of the integral to be u, and the remaining part to be dv. Differentiate u to get du. Integrate dv to get v. Replace u dv with uv vdu. Hope/check that the new integral is easier to evaluate.

5 Integration By Parts - Example 1-1 Example: udv = uv vdu Use integration by parts to evaluate xe 4x dx.

6 Integration By Parts - Example 1-2 Verify that your anti-derivative is correct.

7 Integration By Parts - Example 2-1 udv = uv vdu Guidelines for selecting u and dv Try to select u and dv so that either u is simpler than u or dv is simpler than dv Ensure you can actually integrate the dv part by itself

8 Integration By Parts - Example 2-2 Example: Consider the integral x cos x dx. Based on the guidelines, what choice of u and dv should you try first? (a) u = x cos(x), dv = dx. (b) u = 1, dv = x cos(x) dx. (c) u = x, dv = cos(x) dx. (d) u = cos(x), dv = x dx.

9 Integration By Parts - Example 2-3 Evaluate the integral x cos x dx.

10 Integration By Parts - Example 2-4 Verify that your anti-derivative is correct.

11 Integration By Parts - Example 3-1 Example: Evaluate the slightly more challenging integral x 2 cos x dx What choice of u and dv would be most likely to be helpful? (a) u = 1, dv = x 2 cos(x) dx. (b) u = x, dv = x cos(x) dx. (c) u = x 2, dv = cos(x)dx. (d) u = x 2 cos(x), dv = dx.

12 Integration By Parts - Example 3-2 x 2 cos x dx

13 Integration By Parts - Example 3-3 x 2 cos x dx

14 Integration By Parts - Definite Integrals - 1 Integration By Parts - Definite Integrals When using integration by parts to evaluate definite integrals, you need to apply the limits of integration to the entire anti-derivative that you find. Example: Evaluate π 0 x sin(4x) dx

15 Integration By Parts - Definite Integrals - 2 Don t forget that dv does not require any other factors besides dx. That can help when there is only a single factor in the integrand. Example: Find 2 1 ln x dx

16 Integration By Parts - Circular Case - 1 There are some classical problems that can be solved by integration by parts, but not in the direct way we have seen so far. Example: Integrate by parts twice to find cos(x)e x dx.

17 Integration By Parts - Circular Case - 2 cos(x)e x dx

18 Applications of Integrals - Average Value - 1 Applications of Integrals - Average Value Example: If the following graph describes the level of CO 2 in the air in a greenhouse over a week, estimate the average level of CO 2 over that period. CO 2 (ppm) Time (Days) Give the units of the average CO 2 level.

19 Applications of Integrals - Average Value - 2 Example: Sketch the graph of f(x) = x 2 from x = 2 to 2, and estimate the average value of f on that interval.

20 Applications of Integrals - Average Value - 3 What makes the single average f value different or distinct from other possible f values? Use this property to find a general expression for the average value of f(x) on the interval x [a, b].

21 Applications of Integrals - Average Value - 4 Average Value of a Function on [a, b] The average value of a function f(x) on the interval [a, b] is given by A = 1 b a b a f(x) dx Interpret this form of the average value graphically.

22 Applications of Integrals - Average Value - 5 Example: Find the exact average of f(x) = x 2 on the interval x [ 2, 2] using this formula.

23 Average Value - Examples 1-1 A = 1 b a b a f(x) dx Example: The temperature in a house is given by H(t) = sin(πt/12), where t is in hours and H is degrees C. Sketch the graph of H(t) from t = 0 to t = 12, then find the average temperature between t = 0 and t = 12.

24 Average Value - Examples 1-2 H(t) = sin(πt/12)

25 Average Value - Examples 2-1 Example: A = 1 b a b a f(x) dx You are told that the average value of f(x) over the interval x [0, 3] is 5. What is the value of 3 0 f(x) dx?

26 Average Value - Examples 2-2 Example: Without computing any integrals, explain why the average value of cos(x) on x [0, π/2] must be greater than 0.5.

27 Average Value - Examples 2-3 Can you make a general statement about the average value of decreasing functions which are concave down?

Examples from Section 7.1: Integration by Parts Page 1

Examples from Section 7.1: Integration by Parts Page 1 Examples from Section 7.: Integration by Parts Page Questions Example Determine x cos x dx. Example e θ cos θ dθ Example You may wonder why we do not add a constant at the point where we integrate for